THE JOURNAL OF CHEMICAL PHYSICS 125, 014704 s2006d
Impact of molecular structure on the lubricant squeeze-out between curved
surfaces with long range elasticity
U. Tartaglinoa!
IFF, FZ-Jülich, D-52425 Jülich, Germany and International School for Advanced Studies (SISSA)
and INFM Democritos National Simulation Center, Via Beirut 2-4, I-34014 Trieste, Italy
I. M. Sivebaek
IFF, FZ-Jülich, D-52425 Jülich, Germany; Novo Nordisk A/S, Research and Development, DK-3400
Hillerød, Denmark; and MEK-Energy, Technical University of Denmark, DK-2800 Lyngby, Denmark
B. N. J. Persson
IFF, FZ-Jülich, D-52425 Jülich, Germany
E. Tosatti
International School for Advanced Studies (SISSA) and INFM Democritos National Simulation Center, Via
Beirut 2-4, I-34014 Trieste, Italy and The Abdus Salam International Centre for Theoretical Physics
(ICTP), P.O. Box 586, I-34014 Trieste, Italy
sReceived 27 March 2006; accepted 9 May 2006; published online 5 July 2006d
The properties of butane sC4H10d lubricants confined between two approaching solids are
investigated by a model that accounts for the curvature and elastic properties of the solid surfaces.
We consider the linear n-butane and the branched isobutane. For the linear molecule, well defined
molecular layers develop in the lubricant film when the width is of the order of a few atomic
diameters. The branched isobutane forms more disordered structures which permit it to stay
liquidlike at smaller surface separations. During squeezing the solvation forces show oscillations
corresponding to the width of a molecule. At low speeds s,0.1 m / sd the last layers of isobutane are
squeezed out before those of n-butane. Since the sinterfaciald squeezing velocity in most practical
applications is very low when the lubricant layer has molecular thickness, one expects n-butane to
be a better boundary lubricant than isobutane. With n-butane possessing a slightly lower viscosity
at high pressures, our result refutes the view that squeeze-out should be harder for higher viscosities;
on the other hand our results are consistent with wear experiments in which n-butane were shown
to protect steel surfaces better than isobutane. © 2006 American Institute of Physics.
fDOI: 10.1063/1.2210008g
I. INTRODUCTION
Modern materials are subject to increasing loads and are
used under still more demanding conditions. This emphasizes the need for a better understanding of friction, lubrication, and wear phenomena.1,2
An example of the above is the fuel lubricated diesel
engine injection pump, capable of delivering in excess of
2000 bars today compared to 500 bars 15 years ago. This increase in injection pressure has ensured higher engine efficiencies and lower pollution levels.
Historically the diesel oil sulfur reduction both in the
1960s and 1990s combined with the high pressures has
played a significant role in injection pump durability. Sulfur,
along with other polarity inducing atoms, is part of polar
species that ensure low wear in boundary lubrication, a predominant regime in pumps lubricated by diesel oil. Today
there exist several accelerated laboratory tests capable of predicting the lubricating abilities of a diesel oil. If a fuel fails
such a test, lubricity additives in small proportions are added
to ensure proper wear resistance.
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The currently used diesel oil wear test in Europe is the
high frequency reciprocating rig sHFRRd which is covered
by several standards.3,4 The principle in the HFRR is the
sliding of a fixed steel ball, loaded with 2 N on a steel disk
for 75 min. The motion is reciprocating with a stroke of
1 mm and a frequency of 50 Hz. As the specimen contact is
fuel lubricated the resulting wear scar diameter on the ball
expresses the lubricity of the tested fuel. The configuration of
the HFRR is shown in Fig. 1.
The HFRR ability of predicting the lifetime of injection
pumps has been questioned when low-viscosity fuels are
tested. It appears that a decrease in fuel viscosity requires an
increase in lubricity ssmaller wear scard to ensure full lifetime of the pumps. This relation between viscosity and lubricity was confirmed as a new fuel appeared in the 1990s:
dimethyl ether sDMEd. This fuel has a viscosity 20 times
lower than that of diesel oil and it was established that an
adequate lubricity level according to the HFRR standards
was not sufficient to protect the pump surfaces.5–7
A study8 has argued that fluid viscosity is only a secondary property in the HFRR. Based on molecular dynamics
calculations it was discovered that the squeeze-out of linear
alkanes from surface contacts is primarily a function of the
125, 014704-1
© 2006 American Institute of Physics
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014704-2
Tartaglino et al.
FIG. 1. The ball on disk configuration in a HFRR wear test.
length of the molecule. Experimental data shown in Fig. 2
demonstrate that branched alkanes perform worse in the
HFRR than their linear isomers although their viscosities are
almost the same.6 This observation confirms that the bulk
viscosity may be a secondary property in boundary lubrication.
The friction and wear properties in boundary lubrication
of lubricants with branched and linear molecules have been
investigated both experimentally9–12 and using theoretical
simulation methods.13–17
Whether linear alkanes or their branched isomers are the
best lubricants according to the cited literature remains uncertain. A number of studies indicate that n-alkanes are the
best lubricants,13,14,16,17 whereas others claim that branched
alkanes perform better.9,11,12,15 One major weakness of the
published studies is that the lubricant squeezing is stopped at
a separation of about 8 Å between the surfaces. At this point
there are still a few monolayers left protecting the surfaces,
and significant wear originating from cold welding should
only appear when the last lubricant layer is squeezed out.18
In the present computer simulation study we investigate
theoretically the lubrication abilities of n- and isobutane extending the squeezing down to a surface separation of 0 Å so
that the important expulsion of the last monolayer is addressed.
FIG. 2. The wear scar diameter for various alkanes as a function of the
molecular length. The squares are from Ref. 19 and the circles from Ref. 8.
sad and scd are for linear alkanes and sbd and sdd for branched isomers.
J. Chem. Phys. 125, 014704 ~2006!
FIG. 3. The simulated sample: two elastic walls with lubricant in between
are pushed together. The sinusoidal profile of the upper surface generates a
line contact in the middle, where lubricant forms a neck and eventually it is
squeezed out by the load. Periodic boundary conditions are applied along x
and y.
II. THE MODEL
Figure 3 shows the geometry for the simulation of the
squeeze-out process. Lubricant molecules are confined between two elastic walls. The lower wall, or substrate, is flat,
while the upper one, called block, has a sinusoidal profile, so
that a contact region appears in the center of the sample. The
liquid lubricant adheres to the walls, forming a neck in the
central region. When the block is pushed towards the substrate, the lubricant has space to expand laterally into the
vapor. Periodic boundary conditions are applied along the x
and y directions. The periodically repeated cell forms a rectangle Lx 3 Ly, with Lx = 416 Å and Ly = 52 Å. The contact
region is not circular, but forms an infinite long strip parallel
to the y axis.
The lubricant molecules are described through the optimized potential for liquid simulation sOPLSd;20,21 this potential is known to provide density and viscosity of hydrocarbons close to the experimental one.
Each butane molecule comprises four units sparticlesd,
each particle corresponding to one chemical group CH3,
CH2, or CH sunited atom modeld. The interaction between
particles of different molecules is described by LennardJones potentials. The intramolecular interactions include two
body forces that keep the bond length C–C close to 1.53 Å,
three body forces imposing a preferred angle of 115° between the carbon atoms, and four body forces favoring a well
defined torsion of the molecules. The four body forces apply
to the sequence of carbon atoms C–C–C–C; thus they are
present only for n-butane.
For the isobutane molecules we introduced also an antiumbrella inversion potential, analogous to what has already
been done by Mondello and Grest22 for branched hydrocarbons; the only difference here being that in the isobutane
molecule it is not possible to distinguish which carbon atom
constitutes the lateral branch and which constitute the backbone of the molecule. Therefore we symmetrized the antiinversion potential.
The interaction between the beads of the lubricant molecule and the walls’ atoms is also given by Lennard-Jones
forces, whose energy parameter e0 = 18.6 meV, as in Ref. 23,
yielding an adsorption energy of about 0.3 eV/ molecule,
comparable to that of butane on Aus111d.24
The elastic energy due to the walls’ deformation is also
involved in the nucleation process that triggers the transition
from n to n − 1 monolayers of lubricant in the confined
regime.25 In principle, we can achieve a realistic description
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014704-3
Impact of molecular structure on the lubricant squeeze-out
of the walls’ elastic response by simulating thick walls comprising many layers. Since we necessitate proper elastic behavior up to wavelengths comparable with the size of the
contact area the walls’ thickness must not be smaller that the
lateral size of the box, Ly = 52 Å. This indeed would imply
too many atoms, slowing down the calculations. Instead we
adopted the model described in Ref. 26: for each wall only
the outermost layer of atoms is considered; these atoms are
connected to a rigid surface with springs that take into account both the compressibility and the shear rigidity of the
wall. Similar springs connect together the square grid of atoms of the wall. The advantage of this approach is double:
with a relatively small number of atoms it is possible to
describe the long range elasticity of the walls and to impose
a curved profile simply by using a curved rigid surface.
The rigid surface connected to the substrate’s atoms is
flat and its position is fixed. The upper rigid surface has the
profile
zsx,yd = z0 +
S
S DD
2px
A
1 − cos
2
Lx
,
s1d
where the amplitude of corrugation A is 20, 40, or 60 Å. The
case A = 20 Å guarantees a larger contact region, but it does
not allow enough empty space sor rather vapor space regiond
for the squeeze-out of the last two layers; it was used only
for comparison with the other cases to ensure that the size of
the contact region is large enough for the nucleation of the
squeeze-out process.
The springs connecting the walls’ atoms to the rigid surfaces simulate the elastic response of a gold film of thickness
50 Å, comparable with the lateral size Ly of the sample.
The substrate consists of 1443 18 atoms in a square lattice with lattice spacing of 2.889 Å; the block layer is made
of 1603 20, atoms with lattice spacing of 2.6 Å. The different lattice spacing is to reduce the commensurability between the walls. Both substrate’s and block’s atoms have the
same mass of gold: 197 amu. In most of the simulations we
employed 2800 lubricant molecules, which deposit on the
two walls forming about two monolayers of adsorbate on
each surface.
J. Chem. Phys. 125, 014704 ~2006!
FIG. 4. The confined regime: the liquid lubricant gets ordered into 2D layers
when it is confined between atomically flat surfaces. The picture shows the
side view of isobutane in the contact region between the two walls. Light
balls represent CH and CH3 groups.
simulated cell. The contact region is about 15% of the whole
cell size, and the real pressure in the center of the contact
area is much larger than the average pressure. Negative pressures observed in some time intervals are due to the attractive capillarity forces of the neck of lubricant. The plotted
curves show that isobutane is squeezed out earlier and at a
lower pressure. This is the typical behavior we observed
when there is more than one monolayer. Conversely the removal of the very last monolayer appears to be easier for
n-butane. As can be observed in Fig. 6, the ejection of the
isobutane not only happens later, but it is slower and a small
group of seven molecules remains trapped at the end. This
suggests that the simulated squeeze-out of the last layer is
strongly affected by dynamical effects, such as the drag friction when the lubricant slides on the solid walls. Thus, even
though the nucleation barrier of the hole may be smaller for
isobutane, the simulated system might not have enough time
to exit its metastable state before the pressure is further increased by the vertical ssqueezingd motion of the block.
The behavior at higher temperatures is qualitatively the
same. At T = 350 K the switch from three to two monolayers
and from two to one monolayer happens first for isobutane,
but the very last layer of isobutane stays longer between the
walls, and five isobutane molecules remain trapped there ssee
Fig. 7d. Another independent simulation at T = 330 K snot
shownd confirms this trend.
III. SIMULATION RESULTS
Two simulations are prepared in parallel for iso- and
n-butane with identical conditions. Initially the whole system
is thermalized and the lubricant adheres to the two walls.
Then the rigid upper surface is moved down towards the
substrate with constant speed, i.e., z0 in Eq. s1d is constrained
to decrease linearly with time. As the lubricant layers on the
two walls get in contact, a neck is formed in the contact area.
The lubricant clearly shows layering both for the n-butane
and for the more disordered isobutane sFig. 4d. The change in
the number of monolayers takes place abruptly, with a sudden decrease of pressure due to the relaxation of the elastic
energy stored in the walls.
Figure 5 shows the behavior of pressure versus time
while squeezing at a speed of 1 m / s and at a temperature of
300 K. The y axis contains a spatial average of the pressure,
that is the vertical force divided by the area LxLy of the
FIG. 5. Squeeze-out of butane at T = 300 K. The pressure drops when the
number of layers switches from 3 to 2 and from 2 to 1. Squeezing speed
= 1 m / s.
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014704-4
Tartaglino et al.
J. Chem. Phys. 125, 014704 ~2006!
FIG. 8. Side and top views of the isobutane monolayer at different times
sand pressuresd. Images refer to the simulation of Fig. 7.
FIG. 6. Top view of the lubricant molecules in the central part of the contact
area immediately before the expulsion of the last monolayer stopd, after
0.5 ns smiddled, and after 1 ns. T = 300 K, squeezing speed= 1 m / s. sBlock
and substrate are parallel to the plane of the figure.d
The transition from two to one isobutane monolayer reveals another interesting feature: the pressure drops down in
two stages, as is shown in the plot of Fig. 7 for time close to
1.2 ns. What happens is clearly illustrated in the snapshots of
Fig. 8: initially there is the squeeze-out of one of the two
layers. The remaining monolayer does not have a preferential
orientation of the molecules parallel to the walls; the plateau
in the pressure versus time graph at t < 1.2 ns is due to this
thick monolayer. Finally the molecules change their orientation forming a thinner monolayer, yielding a smaller second
pressure drop. This kind of two-stage squeezing of the second layer has been observed in other simulations too, but
only for isobutane. It is, in fact, a steric effect due to the
shape of the molecule. Similar pressure-induced phase transitions have been observed in other computer simulations of
squeeze-out. Thus, in Ref. 27 it was shown that Xe atoms
FIG. 7. Pressure vs time during the complete squeeze-out of three monolayers at T = 350 K, squeezing speed= 1 m / s. The pictures show the lubricant while the last monolayer is being removed. Corrugation of the upper
profile: A = 60 Å; contact area: 10% of the cell’s area.
between two solid surfaces exhibited triangular bilayers at
low squeezing pressure, which abruptly transformed to
fccs100d layers parallel to the solid surfaces when the pressure increased. This transition allowed the surfaces to move
closer to each other which released elastic energy.
The simulations at lower squeezing speed give clear evidence that the relatively higher lubricity of the last layer of
isobutane is a kinetic effect. Thus, at T = 300 K and for the
compression speed of 1 m / s sFig. 6d n-butane is removed
first, while when the speed is 0.1 m / s the removal of the two
lubricants happens almost at the same time snot shownd. Finally when the squeezing is carried at the speed of 0.03 m / s
isobutane is removed first and at a lower pressure sFig. 9d. In
this case there is still some small island of molecules
trapped, shown in the picture, confirming once more that the
lateral sliding of isobutane is slower.
A further evidence that the removal of the last monolayer is influenced by kinetic effects comes from the squeez-
FIG. 9. Last monolayer squeeze-out at low speed. T = 300 K, squeezing
speed= 0.03 m / s. Isobutane is removed earlier at a lower pressure, but the
cluster of six molecules shown in the picture remains trapped up to t
= 3.6 ns. Below: the average position of the upper wall’s atoms vs time.
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014704-5
Impact of molecular structure on the lubricant squeeze-out
FIG. 10. Pressure required to squeeze-out the last lubricant layer while
pushing at the speed of 0.03 m / s. Data refer to the spatial average over the
whole simulation cell. The pressure in the center of the contact area is much
larger. Each point is the result of a single independent simulation, requiring
about two months of CPU time on a standard PC.
ing pressure versus pushing speed vz: keeping all the other
conditions identical, we observed that the pressure at which
the last layer is removed does decrease when the block is
pushed down slowly. For example, for isobutane at T
= 300 K we obtained the following pressures: 0.340 GPa
when vz = 1 m / s, 0.257 GPa when vz = 0.1 m / s, and
0.239 GPa when vz = 0.03 m / s sas usual, these are spatial
averages, the true pressures in the contact area being much
higherd. Similarly, the corresponding simulations for
n-butane gave 0.321, 0.252, and 0.263 GPa, respectively.
The lubricant is able to nucleate the hole for the squeeze-out
at the lowest of these pressures, but at high pushing speed
there is not enough time to exploit this possibility before the
pressure is further increased.
It is interesting to observe what happens when the temperature is changed, particularly considering that in reality
the microcontacts where the squeeze-out happens are likely
to be much warmer than the environment, at least during
sliding. The temperature is likely to speed up the mobility of
the molecules, reducing the influence of the drag force of the
walls. Moreover the larger thermal fluctuations should
strongly favor the nucleation of the critical hole in the lubricant layer. Both effects tend to reduce the pressure needed to
squeeze the lubricant. This is indeed the trend which we
found and that is observed in Fig. 10. On the other hand, any
difference DG between n-butane and isobutane in the free
energy barriers for nucleation of squeeze-out should become
less important as the temperature increases, as the ratio between the nucleation probabilities is mainly affected by the
Boltzmann factor exps−bDGd, which goes to 1 when T → `.
Unfortunately the low temperature conditions are hard to
analyze through simulations: the randomness of the single
squeeze-out event is enhanced by the strong decrease of the
Boltzmann factor. Actually the transition time for a nucleating process has an exponential distribution, where the standard deviation is equal to the average value. Thereafter the
data of a single simulation become no more reliable and only
an average over many independent simulations—not easily
affordable due to computer time limitations—would show
J. Chem. Phys. 125, 014704 ~2006!
the real trend. We do not have enough statistics in Fig. 10 for
a reliable comparison of the behaviors of n-butane and isobutane below room temperature.
Finally we consider the effect of lateral sliding of the
walls with respect to each other. We ran some simulations
snot shownd at different temperatures with squeezing speed
and lateral speed both equal to 1 m / s, and we observed that
the removal of the last monolayer happens almost simultaneously at the same pressure. The relative motion of the
walls tends to take away the effect of the drag between walls
and lubricant, which is indeed responsible of the higher lubricity of isobutane when squeezing fast. This is an important remark, since in many practical situations there is lateral
sliding.
The significance of the temperature for the squeeze-out
of the two different butanes is of utmost importance for future wear experiments. The results in Fig. 2 are all obtained
at 298 K. At this temperature and at the given sliding conditions it seems that isobutane is more easily squeezed out
from the ball-disk contact than n-butane as the resulting wear
is slightly lower for the latter one. In real injection pumps the
temperature could be very low sat start-upd and very high
srunning conditionsd. The present simulation results imply
that wear protecting properties of different molecules in the
fuel may not react the same way to changes in temperature
and sliding conditions. This means that a wear test may not
reflect reality unless test conditions such as temperature and
sliding velocity are varied. This new aspect of wear experiments should be looked into in the near future.
IV. SUMMARY AND CONCLUSIONS
We have studied the properties of butane sC4H10d lubricants confined between two approaching solids using a
model that accounts for the curvature and elastic properties
of the solid surfaces. We considered both linear n-butane and
branched isobutane. In the case of the linear molecule well
defined molecular layers develop in the lubricant film when
the width is of the order of a few atomic diameters. The
branched isobutane forms more disordered structures which
permit it to stay liquidlike at smaller surface separations.
During squeezing the solvation forces show oscillations
corresponding to the width of a molecule. At low speeds
s,0.1 m / sd the last layers of isobutane are squeezed out
before those of n-butane. Since the sinterfaciald squeezing
velocity in most practical applications is very low when the
lubricant layer has molecular thickness, one expect n-butane
to be a better boundary lubricant than isobutane. This is consistent with wear experiments in which n-butane was shown
to protect steel surfaces better than isobutane. On the other
hand, with n-butane possessing lower viscosity at high pressures, our result refutes the view that squeeze-out should be
harder for higher viscosities. At high squeezing velocity, the
squeeze-out of the last monolayer of isobutane occurs at
higher pressures than for n-butane. We interpret this as a
kinetic effect resulting from the lateral corrugation barrier
experienced by the molecules. As the temperature increases,
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014704-6
Tartaglino et al.
the squeeze-out occurs at lower applied pressure for both
n-butane and isobutane. This is the expected result based on
the nucleation theory of squeeze-out.25,27
ACKNOWLEDGMENTS
Two of the authors sU.T. and I.M.S.d acknowledge support from IFF, FZ-Jülich and the hospitality and help of the
staff during their research visits. One of the authors sI.M.S.d
also acknowledges financial support from the European
project AFFORHD. In Trieste, this work was also sponsored
by MIUR COFIN No. 2003028141-007, MIUR COFIN No.
2004028238-002, MIUR FIRB RBAU017S8 R004, and
MIUR FIRB RBAU01LX5H.
1
B. N. J. Persson, Sliding Friction: Physical Principles and Applictions
sSpringer, Heidelberg, 2000d.
2
J. N. Israelachvili, Intermolecular and Surface Forces sAcademic, London, 1995d.
3
Annual Book of ASTM Standards: D6079-97 5.03, 1355 s1998d.
4
ISO Standards ISO 12156-1 and 2 s1997d.
5
I. M. Sivebaek and S. C. Sorenson, Soc. Automot. Eng. fSpec. Publ.g
SAE Tech. Paper 2000–01–2970 s2000d.
6
I. M. Sivebaek, S. C. Sorenson, and J. Jakobsen, Tribol. Lett. sto be
publishedd.
7
I. M. Sivebaek, S. C. Sorenson, and J. Jakobsen, Soc. Automot. Eng.
fSpec. Publ.g SAE Tech. Paper 2001–01–2013 s2001d.
J. Chem. Phys. 125, 014704 ~2006!
8
I. M. Sivebaek, V. N. Samoilov, and B. N. J. Persson, J. Chem. Phys.
119, 2314 s2003d.
9
H. K. Christenson, J. Chem. Phys. 78, 6906 s1983d.
10
H. K. Christeenson, R. G. Horn, and J. N. Israelachvili, J. Colloid Interface Sci. 88, 79 s1982d.
11
J. Israelachvili, S. J. Kott, M. L. Gee, and T. A. Witten, Macromolecules
22, 4247 s1989d.
12
M. L. Gee, P. M. McGuiggan, and J. N. Israelachvili, J. Chem. Phys. 93,
1895 s1990d.
13
Y. Wang, K. Hill, and J. G. Harris, J. Chem. Phys. 100, 3276 s1994d.
14
M. Dijkstra, J. Chem. Phys. 107, 3277 s1997d.
15
H. Tamura, M. Yoshida, K. Kusakabe, C. Young-Mo, R. Miura, M. Kubo,
K. Teraishi, A. Chatterjee, and A. Miyamoto, Langmuir 15, 7816 s1999d.
16
J. C. Wang and K. A. Fichthorn, J. Chem. Phys. 116, 410 s2002d.
17
S. T. Cui, P. T. Cummings, and H. D. Cochran, J. Chem. Phys. 114, 6464
s2001d.
18
A. M. Homola, J. N. Israelachvili, P. M. McGuiggan, and M. L. Gee,
Wear 136, 65 s1990d.
19
D. P. Wei, H. A. Spikes, and S. Korcek, Tribol. Trans. 42, 813 s1999d.
20
W. I. Jorgensen, J. D. Madura, and C. J. Swenson, J. Am. Chem. Soc.
106, 6638 s1984d.
21
D. K. Dysthe, A. H. Fuchs, and B. Rousseau, J. Chem. Phys. 112, 7581
s2000d.
22
M. Mondello and G. S. Grest, J. Chem. Phys. 103, 7156 s1995d.
23
T. K. Xia, J. Ouyang, M. W. Ribarsky, and U. Landman, Phys. Rev. Lett.
69, 1967 s1992d.
24
S. M. Wetterer, D. J. Lavrich, T. Cummings, S. L. Bernasek, and G.
Scoles, J. Phys. Chem. B 102, 9266 s1998d.
25
B. N. J. Persson and E. Tosatti, Phys. Rev. B 50, 5590 s1994d.
26
B. N. J. Persson and P. Ballone, J. Chem. Phys. 112, 9524 s2000d.
27
B. N. J. Persson and F. Mugele, J. Phys.: Condens. Matter 16, R295
s2004d.
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